Hi, my name is Murray. I am a grade 10 hight school student. I'm working on a complicated proof and i need the equation for a line in 3 dimensions. I haven't got very far. Could you please give me some hints. thanx. Hi Murray,By "the" equation of a line, what is usually meant is the "parametric" equation: Given two points, say (1,6,3) and (8,2,7), you you take their difference: This result (7,4,4) you call the "direction vector" of the line passing through these two points. The parametric equation of the line is then Here t is called the "parameter": imputing a value t gives you a point on the line. For instance with t=0, you get the point (1,6,3), with t=1 you get (8,2,7), with t=5/2 you get (37/2,4,13), and so on. You could do the same in two dimensions: With two points (1,6), (8,2), you find the direction vector (8,2)  (1,6) = (7,4), and you get the parametric equation To get the equation that you are familiar with, you take the two coordinate equations: the first coordinate is x = 1 + 7t, and the second is y = 6  4t, and you combine them to cancel out the t. For instance, 4 times the first equation plus 7 times the second equation gives you 4x + 7y = 4 + 28t + 42  28t = 46, so you get 4x + 7y = 46, which is the "implicit" equation of the line. It tells you, for instance, that the point (8,2) is on the line, because 4*8 + 7*2 = 46, while (9,1) is not on the line, because 4*9 + 7*1 is not 46. Now, to return to 3 dimensions: Lines in 3 dimensions also have implicit equations, but now it takes two equations to characterise a line. One equation alone would characterise a plane, not a line. To find these equations, you proceed as above: the coordinate equations for are x = 1 + 7t, y = 6  4t, z = 3 + 4t. As mentionned above the two first equations combine to give 4x + 7y = 46. Now adding the second equation to the third, you get y + z = 9; which is your second equation. Together, these two equations characterise the line, it is the line of intersection of the two planes. They tell you that the point (15,2,11) is on the line, because 4*15 + 7*(2) = 46 and (2) + 11 = 9, while (8,2,4) is not on the line, because 2 + 4 is not 9 (even though the first equation is satisfied). I hope this answers your question. Good luck with your proof.
